With this circuit you think you will feed the USB with a max of 0.59A or 0.59*5.4W = just over 3 watts.
In fact you probably don't get that much since 7806 should regulate at 6V, so it is probably only being fed 6-7 volts and not 8V. 8V is the highest voltage you will see from the cell and only with no load in bright sunlight.
Likewise the 0.59A rating is only achieved at zero volts, i.e. into a short circuit.
This is the way these panels are specified. There is a published curve available from the manufacturer that plots Vols and current vs incident light.
The maximum voltage or current is only achieved at maximum incident light on the panel in noonday sun and at a voltage less than the maximum. Check with a voltmeter and ammeter to see what is actually happening.
Additionally, the 7806 is dropping whatever the difference is between the output of the solar panel and 5.4V as heat. You are likely overheating it and it will drop the current supply to avoid being destroyed.
if 7806 gets hot, try fitting a heatsink.
You can also try a small switching power supply from eBay, DX etc. This is a ready made buck converter. They are reasonably priced and will give 2-5 amps. They are also much more efficient in this type of application, providing approx.
watts out =approx 0.9 * watts in
provided they are operated within their specifications.
In this situation, a small car charger that fits into a cigarette lighter may work 6-8V you get is probably not sufficient. Use two panels in series